This paper proposes a reformulation of the scenario-based two-stage unit commitment problem under uncertainty that allows finding unit-commitment plans that perform reasonably well both in expectation and for the worst case realization of the uncertainties. The proposed reformulation is based on partitioning the sample space of the uncertain factors by clustering the scenarios that approximate their probability distributions. It is, furthermore, very amenable to decomposition and parallelization using a column-and-constraint generation procedure.
Day-ahead scheduling of electricity generation or unit commitment is an important and challenging optimization problem in power systems. Variability in net load arising from the increasing penetration of renewable technologies have motivated study of various classes of stochastic unit commitment models. In two-stage models, the generation schedule for the entire day is fixed while the dispatch is adapted to the uncertainty, whereas in multi-stage models the generation schedule is also allowed to dynamically adapt to the uncertainty realization. Multi-stage models provide more flexibility in th…
In this paper we introduce a continuous time multi stage stochastic optimization for scheduling generating units, their commitment, reserve capacities and their continuous time generation profiles in the day-ahead wholesale electricity market. Our formulation approximates the solution of a variational problem, in which the balance, generation capacity and ramping constraints are in continuous time. Due to the greater accuracy of our representation of ramping events this approach improves the system reliability and lowers the real-time cost.
As net-load becomes less predictable there is a lot of pressure in changing decision models for power markets such that they account explicitly for future scenarios in making commitment decisions. This paper proposes to make commitment decisions using a dynamic multistage stochastic unit commitment formulation over a cohesive horizon that leverages a state-space model for the commitment variables. We study the problem of constructing scenario tree approximations for both original and residual stochastic process and evaluate our algorithms on scenario tree libraries derived from real net-load d…
We consider the sum of the incurred start-up costs of a single unit in a Unit Commitment problem. Our major result is a correspondence between the facets of its epigraph and some binary trees for concave start-up cost functions CU, which is bijective if CU is strictly concave. We derive an exponential H-representation of this epigraph, and provide an exact linear separation algorithm. These results significantly reduce the integrality gap of the Mixed Integer formulation of a Unit Commitment Problem compared to current literature.
The short-term operation of a power system is usually planned by solving a day-ahead unit commitment problem. Due to historical reasons, the commitment of the power generating units is decided over a time horizon typically consisting of the 24 hourly periods of a day. In this paper, we show that, as a result of the increasing penetration of intermittent renewable generation, this somewhat arbitrary and artificial division of time may prove to be significantly suboptimal and counterproductive. Instead, we propose a time-adaptive day-ahead unit commitment formulation that better captures the net…
This paper proposes a sequential convex relaxation method for obtaining feasible and near-globally optimal solutions for unit commitment (UC) with AC transmission constraints. First, we develop a second-order cone programming (SOCP) relaxation for AC unit commitment. To ensure that the resulting solutions are feasible for the original non-convex problem, we incorporate penalty terms into the objective of the proposed SOCP relaxation. We generalize our penalization method to a sequential algorithm which starts from an initial point (not necessarily feasible) and leads to feasible and near-optim…
We study the multi-stage stochastic unit commitment problem in which commitment and generation decisions can be made and adjusted in each time period. We formulate this problem as a Markov decision process, which is "weakly-coupled" in the sense that if the demand constraint is relaxed, the problem decomposes into a separate, low-dimensional, Markov decision process for each generator. We demonstrate how the dual approximate dynamic programming method of Barty, Carpentier, and Girardeau (RAIRO Operations Research, 44:167-183, 2010) can be adapted to obtain bounds and a policy for this problem.…
Starting from a basic Unit Commitment problem as published in [Carrión2006], we develop a fully functional, running implementation for Xpress Mosel, useable in power market modeling. The constraints of the model are discussed in details and solutions to left open implementation problems are presented, guidelines on how to handle data input (from both, databases and Excel), as well as data output (to Excel) are given.
Optimization models have been broadly used within side the energy industry as useful decision-making systems for scheduling and dispatching electric powered energy resources; this is applied in a system called unit commitment (UC). Unit Commitment seeks the maximum price adequate generator commitment scheme for an electric powered energy device to satisfy a certain demand, at the same time as fulfilling the operational constraints on transmission models and computational resources. Taking into account risk variability in those processes and checking out their comparative overall performance fo…
This paper presents an improved mixed-integer model for the Thermal Unit Commitment Problem. By introducing new variables for the temperature of each thermal unit, the off-time-dependent start-up costs are modeled accurately and with a lower integrality gap than state-of-the-art formulations. This new approach significantly improves computational efficiency compared to existing formulations, even if they only model a rough approximation of the start-up costs. Our findings were validated on real-world test cases using CPLEX.
With the integration of large-scale renewable energy sources to power systems, many optimization methods have been applied to solve the stochastic/uncertain transmission-constrained unit commitment (TCUC) problem. Among all methods, two-stage and multi-stage robust optimization-based methods are the most widely adopted ones. In the two-stage methods, nonanticipativity of economic dispatch (ED) decisions are not considered. While in multi-stage methods, explicit decision rules (for example, affine decision rules) are usually adopted to guarantee nonanticipativity of ED decisions. With explicit …
We propose a novel global solution algorithm for the network-constrained unit commitment problem incorporating a nonlinear alternating current model of the transmission network, which is a nonconvex mixed-integer nonlinear programming (MINLP) problem. Our algorithm is based on the multi-tree global optimization methodology, which iterates between a mixed-integer lower-bounding problem and a nonlinear upper-bounding problem. We exploit the mathematical structure of the unit commitment problem with AC power flow constraints (UC-AC) and leverage optimization-based bounds tightening, second-order …
In this paper, we present and study a robust unit commitment model and some variants that consider complex temperature and demand uncertainties. Since there is a strong relationship among the efficiency of gas generators, demand, and temperature in practical systems, our robust models have both left- and right-hand-side (LHS and RHS, respectively) uncertainties. Unlike many existing robust models with RHS uncertainty only, the introduction of LHS uncertainty imposes a huge challenge in computing robust solutions. For those complex formulations, we analyze their structures, derive important pro…
Recently increasing penetration of renewable energy generation brings challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, a unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this paper, we focus on deriving polynomial time algorithms for the unit commitment problems with general convex cost function and piecewise linear cost functio…
In this paper, we study unit commitment (UC) problems considering the uncertainty of load and wind power generation. UC problem is formulated as a chance-constrained two-stage stochastic programming problem where the chance constraint is used to restrict the probability of load imbalance. In addition to the conventional mixed integer linear programming formulation using Big-M, we present the bilinear mixed integer formulation of chance constraint, and then derive its linear counterpart using McCormick linearization method. Then, we develop a bilinear variant of Benders decomposition method, wh…
In this paper we present a formulation of the unit commitment problem with AC power flow constraints. It is solved by a Benders decomposition in which the unit commitment master problem is formulated as a mixed-integer problem with linearization of the power generation constraints for improved convergence. Semidefinite programming relaxation of the rectangular AC optimal power flow is used in the subproblem, providing somewhat conservative cuts. Numerical case studies, including a 6-bus and the IEEE 118-bus network, are provided to test the effectiveness of our proposal. We show in our numeric…
Hydro unit commitment is the problem of maximizing water use efficiency while minimizing start-up costs in the daily operation of multiple hydro plants, subject to constraints on short-term reservoir operation, and long-term goals. A low-rank rounding heuristic is presented for the semidefinite relaxation of the mixed-integer quadratic-constrained formulation of this problem. In addition to limits on reservoir and generator operation, transmission constraints are represented by an approximate AC power flow model. In our proposed method, the mathematical program is equivalently formulated as a …
This paper proposes a Clustered Unit Commitment (CUC) formulation to accurately model flexibility requirements such as ramping, reserve, and startup/shutdown constraints. The CUC is commonly applied in large and long-term planning models to approximate the units' operational flexibility in power systems due to its computational advantages. However, the classic CUC intrinsically and hiddenly overestimates the individual unit's flexibility, thus being unable to replicate the result of the individual UC. This paper then present a set of constraints to correctly represent the units' hidden flexibi…
In this paper, we consider the polyhedral structure of the unit commitment polytope. In particular, we provide the convex hull results for the problem under the following different settings: 1) the convex hulls for the integrated minimum-up/-down time and ramping polytope under the general $T$ time period setting in which the ramping rate equals to the gap between the generation upper and lower bounds and equals to half of the gap between the generation upper and lower bounds, respectively, 2) the convex hull for the integrated minimum-up/-down time and ramping-up polytope for the problem unde…
The discrete unit commitment problem with min-stop ramping constraints optimizes the daily production of thermal power plants (coal, gas, fuel units). For this problem, compact Integer Linear Programming (ILP) formulations have been designed to solve exactly small instances and heuristically real-size instances. This paper investigates whether Dantzig-Wolfe reformulation allows to improve the previous exact method and matheuristics. The extended ILP formulation is presented with the column generation algorithm to solve its linear relaxation. The experimental results show that the Dantzig-Wolfe…
Security-constrained unit commitment (SCUC) model is used for power system day-ahead scheduling. However, current SCUC model uses a static network to deliver power and meet demand optimally. A dynamic network can provide a lower optimal cost and alleviate network congestion. However, due to the computational complexity and the lack of effective algorithms, network reconfiguration has not been included in the SCUC model yet. This paper presents a novel approach to handle the computational complexity in security-constrained unit commitment (SCUC) with corrective network reconfiguration (CNR) whi…
This paper proposes a global optimization method for it ensures finding good solutions while solving the unit commitment (UC) problem with carbon emission trading (CET). This method con-sists of two parts. In the first part, a sequence of linear inte-ger-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original mixed integer nonlinear pro-gramming problem (MINLP) model. In the second part, the algo-rithm introduces the idea of center-cut so that it can quickly find good solutions. The approach tested on 10 test instances with units ranging from 35 to 15…
The N-1-1 contingency criterion considers the con- secutive loss of two components in a power system, with intervening time for system adjustments. In this paper, we consider the problem of optimizing generation unit commitment (UC) while ensuring N-1-1 security. Due to the coupling of time periods associated with consecutive component losses, the resulting problem is a very large-scale mixed-integer linear optimization model. For efficient solution, we introduce a novel branch-and-cut algorithm using a temporally decomposed bilevel separation oracle. The model and algorithm are assessed using…
Vehicle-to-grid (V2G) technology has drawn great interest in the recent years and its efficiency depends on scheduling of charging process of plug-in electric vehicles (PEVs) as small portable power plants in smart parking lots. On the other hand, active shift from centralized to decentralized power generation and environmental concerns have caused an increase in the utilization of combined heat and power (CHP) units in power systems. The goal of this study is to develop and simulate a novel approach for combined heat and power unit commitment with PEVs for cost reduction in electric power sys…
To contribute to a smart grid, the unit commitment and dispatch (UCD) is a very important problem to be revisited in the future microgrid environment. In this paper, the UCD problem is studied by taking account of both traditional thermal units and distributed generation units (DGs) while considering demand response (DR) and carbon emissions trading (CET). Then, the UCD problem is reformulated as the optimal switching problem of a hybrid system. The reformulation allows convenient handling of time-dependent start-up cost for dynamic programming (DP). Afterwards, a closed-loop hierarchical oper…
Recent changes in the fuel mix for electricity generation and, in particular, the increase in Gas-Fueled Power Plants (GFPP), have created significant interdependencies between the electrical power and natural gas transmission systems. However, despite their physical and economic couplings, these networks are still operated independently, with asynchronous market mechanisms. This mode of operation may lead to significant economic and reliability risks in congested environments as revealed by the 2014 polar vortex event experienced by the northeastern United States. To mitigate these risks, whi…
Unified Power Flow Controller (UPFC) is recognized as the most powerful flexible AC transmission systems (FACTS) device for power system operation. This paper addresses how UPFC explores the transmission flexibility and facilitates the integration of uncertain and volatile wind power generation. To this end, a comprehensive unit commitment (UC) model with UPFC and uncertain wind power generation is proposed. Then, some metrics are introduced to evaluate the impacts of UPFC on the reliability, security and economy of power system operation. Further, different dispatch strategies of UPFC are com…
To capture the stochastic characteristics of renewable energy generation output, the chance-constrained unit commitment (CCUC) model is widely used. Conventionally, analytical solution for CCUC is usually based on simplified probability assumption or neglecting some operational constraints, otherwise scenar-io-based methods are used to approximate probability with heavy computation burden. In this paper, Gaussian mixture model (GMM) is employed to characterize the correlation between wind farms and probability distribution of their forecast errors. In our model, chance constraints including re…
The blast wave of machine learning and artificial intelligence has also reached the power systems community, and amid the frenzy of methods and black-box tools that have been left in its wake, it is sometimes difficult to perceive a glimmer of Occam's razor principle. In this letter, we use the unit commitment problem (UCP), an NP-hard mathematical program that is fundamental to power system operations, to show that simplicity must guide any strategy to solve it, in particular those that are based on learning from past UCP instances. To this end, we apply a naive algorithm to produce candidate…
The global climate change creates a dire need to mitigate greenhouse gas (GHG) emissions from thermal generation resources (TGRs). While microgrids are instrumental in enabling the deeper penetration of renewable resources, the short-term planning of microgrids needs to explicitly assess the full range of impact of GHG emissions. To this end, we propose a novel unit commitment (UC) approach, which enables the representation of GHG emissions from TGRs, the specification of GHG emission constraints, and the ex-ante evaluation of carbon tax payment with all other costs and benefits. We quantify t…
This paper proposes a data-driven version of the Benders decomposition algorithm applied to the stochastic unit commitment (SUC) problem. The proposed methodology aims at finding a trade-off between the size of the Benders master problem and the number of iterations until convergence. Using clustering techniques, we exploit the information contained in the Lagrange multipliers of the Benders subproblems in order to aggregate the optimality cuts, without compromising the critical information that is passed to the master problem. In addition, we develop an outer parallelization scheme that finds…
Two-stage robust unit commitment (RUC) models have been widely used for day-ahead energy and reserve scheduling under high renewable integration. The current state of the art relies on budget-constrained polyhedral uncertainty sets to control the conservativeness of the solutions. The associated lack of interpretability and parameter specification procedures, as well as the high computational burden exhibited by available exact solution techniques call for new approaches. In this work, we use an alternative scenario-based framework whereby uncertain renewable generation is characterized by a p…
The deepening penetration of renewable resources into power systems entails great difficulties that have not been surmounted satisfactorily. An issue that merits special attention is the short-term planning of power systems under net load uncertainty. To this end, we work out a distributionally robust unit commitment methodology that expressly assesses the uncertainty associated with net load. The principal strength of the proposed methodology lies in its ability to represent the probabilistic nature of net load without having to set forth its probability distribution. This strength is brought…
The reduced level of system inertia in low-carbon power grids increases the need for alternative frequency services. However, simultaneously optimising the provision of these services in the scheduling process, subject to significant uncertainty, is a complex task given the challenge of linking the steady-state optimisation with frequency dynamics. This paper proposes a novel frequency-constrained Stochastic Unit Commitment (SUC) model which, for the first time, co-optimises energy production along with the provision of synchronised and synthetic inertia, Enhanced Frequency Response (EFR), Pri…
Valve-point loading affects the input-output characteristics of generating units, bringing the fuel costs nonlinear and nonsmooth. This has been considered in the solution of load dispatch problems, but not in the planning phase of unit commitment. This paper presents a mathematical optimization model for the thermal unit commitment problem considering valve-point loading. The formulation is based on a careful linearization of the fuel cost function, which is modeled with great detail on power regions being used in the current solution, and roughly on other regions. A set of benchmark instance…
This paper addresses two vital issues which are barely discussed in the literature on robust unit commitment (RUC): 1) how much the potential operational loss could be if the realization of uncertainty is beyond the prescribed uncertainty set; 2) how large the prescribed uncertainty set should be when it is used for RUC decision making. In this regard, a robust risk-constrained unit commitment (RRUC) formulation is proposed to cope with large-scale volatile and uncertain wind generation. Differing from existing RUC formulations, the wind generation uncertainty set in RRUC is adjustable via cho…
To take unit commitment (UC) decisions under uncertain net load, most studies utilize a stochastic UC (SUC) model that adopts a one-size-fits-all representation of uncertainty. Disregarding contextual information such as weather forecasts and temporal information, these models are typically plagued by a poor out-of-sample performance. To effectively exploit contextual information, in this paper, we formulate a conditional SUC problem that is solved given a covariate observation. The presented problem relies on the true conditional distribution of net load and so cannot be solved in practice. T…
In this paper, the Unit Commitment (UC) problem in a power network with low levels of rotational inertia is studied. Frequency-related constraints, namely the limitation on Rate-of-Change-of-Frequency (RoCoF), frequency nadir and steady-state frequency error, are derived from a uniform system frequency response model and included into a stochastic UC that accounts for wind power and equipment contingency uncertainties using a scenario-tree approach. In contrast to the linear RoCoF and steady-state frequency error constraints, the nadir constraint is highly nonlinear. To preserve the mixed-inte…
With rapid integration of power sources with uncertainty, robustness must be carefully considered in the transmission constrained unit commitment (TCUC) problem. The overall computational complexity of the robust TCUC methods is closely related to the vertex number of the uncertainty set. The vertex number is further associated with 1) the period number in the scheduling horizon as well as 2) the number of nodes with uncertain injections. In this paper, a column merging method (CMM) is proposed to reduce the computation burden by merging the uncertain nodes, while still guar-anteeing the robus…
The joint management of heat and power systems is believed to be key to the integration of renewables into energy systems with a large penetration of district heating. Determining the day-ahead unit commitment and production schedules for these systems is an optimization problem subject to uncertainty stemming from the unpredictability of demand and prices for heat and electricity. Furthermore, owing to the dynamic features of production and heat storage units as well as to the length and granularity of the optimization horizon (e.g., one whole day with hourly resolution), this problem is in e…
In the face of extreme events, e.g., hurricanes, the transmission systems, especially the transmission lines, are affected across time and space. To mitigate these impacts on the day-ahead market from a probabilistic perspective, a resilient unit commitment (UC) problem is formulated as a two-stage robust optimization (RO) problem. In the first stage, the status, energy, and reserves of generators are pre-scheduled to minimize the operational cost, responding to the worst line failure scenario in the operating day. The failure, operation status, and repair of transmission lines are depicted by…
The deep penetration of wind and solar power is a critical component of the future power grid. However, the intermittency and stochasticity of these renewable resources bring significant challenges to the reliable and economic operation of power systems. Motivated by these challenges, we present a multistage adaptive robust optimization model for the unit commitment (UC) problem, which models the sequential nature of the dispatch process and utilizes a new type of dynamic uncertainty sets to capture the temporal and spatial correlations of wind and solar power. The model also considers the ope…
The thermal unit commitment (UC) problem has historically been formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale systems. The tighter characteristic reduces the search space, therefore, as a natural consequence, significantly reduces the computational burden. In literatures, many tightened formulations for a single unit with parts of constraints were reported without presenting explicitly how they were derived. In this paper, a systematic approach is developed to formulate tight formulations. The idea is to use more …
The transmission-constrained unit commitment (TC-UC) problem is one of the most relevant problems solved by independent system operators for the daily operation of power systems. Given its computational complexity, this problem is usually not solved to global optimality for real-size power systems. In this paper, we propose a data-driven method that leverages historical information to screen out network constraints in the TC-UC problem. First, past data on demand and renewable generation throughout the network are used to learn the congestion status of transmission lines. Then, we infer the li…
